Complex Numbers Calculator

Perform complex number operations step by step with explanations and visual analysis.

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Operation: Complex Addition

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About Complex Numbers

What are Complex Numbers?

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1). They extend the real number system and are essential in many areas of mathematics and physics.

Common Complex Number Operations

  1. Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
  2. Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
  3. Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
  4. Division: Use the conjugate to rationalize the denominator
  5. Conjugate: The conjugate of a + bi is a - bi
  6. Modulus: |a + bi| = √(a² + b²)

Applications in Real Life

  • Electrical engineering and circuit analysis
  • Signal processing and communications
  • Quantum mechanics and physics
  • Control systems and automation

Frequently Asked Questions

What is the imaginary unit i?

The imaginary unit i is defined as i² = -1. It allows us to work with square roots of negative numbers.

How do I plot complex numbers?

Complex numbers can be plotted on the complex plane, where the real part is on the x-axis and the imaginary part is on the y-axis.